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1. The set Fred (S) of reduced words over S ∪ S forms a group with respect to the composition Fred (S) × Fred (S) −→ Fred (S) given by (s1 . . sn , sn+1 . . sm ) −→ (s1 . . sn−r sn+1+r . . sn+m ), where s1 . . sn and sn+1 . . sm are elements of Fred (S), and r := max k ∈ {0, . . ,k−1} sn−j = sn+1+j ∨ sn−j = sn+1+j 2. The group Fred (S) is freely generated by S. Proof. Ad. 1. The above composition is well-defined because if two reduced words are composed, then the composed word is reduced by construction.

Then we write Fn for “the” group freely generated by S, and call Fn the free group of rank n. 11. 12), even free subgroups of (countably) infinite rank. 3). 12. , a group G is finitely generated if and only if there exists a finitely generated free group F and a surjective group homomorphism F −→ G. 2. Groups via generators and relations 25 Proof. , the image of a finite generating set is a finite generating set of the quotient). Conversely, let G be a finitely generated group, say generated by the finite set S ⊂ G.

If ϕ(1) has interesting eigenvalues. • Let G be a group. Then the lamplighter group over G is the semi-direct product group ϕ Z, where Z acts on the product ZG Z G by shifting the factors: ϕ : Z −→ Aut G Z z −→ (gn )n∈Z → (gn+z )n∈Z • More generally, the wreath product of two groups G and H is the semidirect product ϕ H, where ϕ is the shift action of H on HG H G. The wreath product of G and H is denoted by G H. 34). 2 Free products and free amalgamated products We now describe a construction that “glues” two groups along a common subgroup.

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